Questions: Divide (m^5+83)/(m^2+m-6)

Solution Steps

To divide the polynomial \(\frac{m^{5}+83}{m^{2}+m-6}\), we can use polynomial long division or synthetic division. However, since the divisor is a quadratic polynomial, polynomial long division is more appropriate. We will divide the terms of the numerator by the leading term of the denominator and subtract the result from the numerator iteratively until we get a remainder.

We start with the numerator and denominator defined as follows:

• Numerator: \( m^5 + 83 \)
• Denominator: \( m^2 + m - 6 \)

We will divide \( m^5 + 83 \) by \( m^2 + m - 6 \) using polynomial long division.

1. Divide the leading term of the numerator \( m^5 \) by the leading term of the denominator \( m^2 \) to get \( m^3 \).

2. Multiply the entire denominator \( m^2 + m - 6 \) by \( m^3 \) to get \( m^5 + m^4 - 6m^3 \).

3. Subtract this result from the numerator: \[ (m^5 + 83) - (m^5 + m^4 - 6m^3) = -m^4 + 6m^3 + 83 \]

4. Repeat the process with the new polynomial \( -m^4 + 6m^3 + 83 \):

   • Divide \( -m^4 \) by \( m^2 \) to get \( -m^2 \).
   • Multiply the denominator by \( -m^2 \) to get \( -m^4 - m^3 + 6m^2 \).
   • Subtract: \[ (-m^4 + 6m^3 + 83) - (-m^4 - m^3 + 6m^2) = 7m^3 - 6m^2 + 83 \]

5. Continue this process until the degree of the remainder is less than the degree of the denominator.

After performing the polynomial long division, we find:

• Quotient: \( m^3 - m^2 + 7m + 6 \)
• Remainder: \( 83 \)

Final Answer